Erdős Problem #1190 — Put … where the maximum is taken over all disjoint systems for which $n_1 > m$.

Erdős Problem #1190 asks for the asymptotic size of the largest reciprocal sum from a disjoint family of residue classes with moduli exceeding $m$. Erdős credited Mirsky and Newman with the basic bound, while the sharp asymptotic is now claimed as a consequence of work on Problem #202.

Known results

• Mirsky and Newman: $psilon_m<1$.
• de la Bretèche, Ford, and Vandehey: $L(m)^{-1+o(1)}<psilon_m<L(m)^{-\sqrt{3}/2+o(1)}$, where $L(m)=\exp\!\left(\u0003\sqrt{\log m\log\log m}\u0003\right)$.
• Erdős reportedly did not know whether $psilon_m\to0$.

April 2026 claimed resolution

GPT-5.4 Pro is credited with implying $psilon_m=L(m)^{-1+o(1)}$ from a claimed sharp result for Problem #202. The claim includes a proposed argument and Lean formalization, but no independent proof or published corroboration was found.

Current status (as of April 2026): The classical bounds are established, but the claimed sharp asymptotic $psilon_m=L(m)^{-1+o(1)}$ remains unverified, so the problem is still open.